三次可平面图与Legendrian曲面理论
Cubic Planar Graphs and Legendrian Surface Theory
摘要：
研究了由三次可平面图确定的勒让德曲面。具有不同色多项式的图确定不是勒让德同位素的表面，从而给出了许多具有相同经典不变量的非同位素勒让德表面的例子。勒让德函数没有精确的拉格朗日填充，但有许多有趣的非精确填充。我们得到这些结果，通过研究三球与微支撑的表面上的层。这种层的模有一个具体的描述方面的图形和一个美丽的嵌入作为一个全纯拉格朗日子流形的辛周期域，拉格朗日已出现在工作的Dimofte-Gabella-Goncharov [DGGo]。我们利用这个结构找到非精确填充的开放Gromov-Witten不变量，遵循Aganagic-Vafa [AV，AV 2]。
英文摘要：
We study Legendrian surfaces determined by cubic planar graphs. Graphs with distinct chromatic polynomials determine surfaces that are not Legendrian isotopic, thus giving many examples of non-isotopic Legendrian surfaces with the same classical invariants. The Legendrians have no exact Lagrangian fillings, but have many interesting non-exact fillings. We obtain these results by studying sheaves on a three-ball with microsupport in the surface. The moduli of such sheaves has a concrete description in terms of the graph and a beautiful embedding as a holomorphic Lagrangian submanifold of a symplectic period domain, a Lagrangian that has appeared in the work of Dimofte-Gabella-Goncharov [DGGo]. We exploit this structure to find conjectural open Gromov-Witten invariants for the non-exact filling, following Aganagic-Vafa [AV, AV2].
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